A^k matrix singularity and (A^k)^-1 = (A^-1)^k

[ramtin]

Homework Statement

Let A be nonsingular. Prove That for any positive integer k , A^k is nonsingular, And (A^k)^-1 = (A^-1)^k.

Homework Equations

The Attempt at a Solution

 

[cipher42]

What have you tried so far?

 

[statdad]

Fill this in: a problem that requires you to prove a simple expression is true for every positive integer k is a good candidate for m ************ ******n

 

[ramtin]

Fill this in: a problem that requires you to prove a simple expression is true for every positive integer k is a good candidate for m ************ ******n

You answer was not complete ...What are the * ?
Please somebody help me!

 

[Office_Shredder]

Start small. Can you prove it's true for k=2? How can you generalize the proof?

 

[ramtin]

Start small. Can you prove it's true for k=2? How can you generalize the proof?

I can't prove it for 2 ,,,don't know How to generalize that

 

[Office_Shredder]

Start by contradiction. If A is nonsingular, then if A² is singular what can we prove about A? Try messing around with the equation A²v = 0

 

[HallsofIvy]

Then what do you know? Under what conditions is a matrix "singular"? What does having an inverse mean?